Goldschmidt tolerance factor

Goldschmidt's tolerance factor is an indicator for the stability and distortion of crystal structures. It was originally only used to describe perovskite structure, but now tolerance factors are also used for ilmenite.

Alternatively the tolerance factor can be used to calculate the compatibility of an ion with a crystal structure.

The first description of the tolerance factor for perovskite was made by Victor Moritz Goldschmidt in 1926.

Mathematical expression

The Goldschmidt tolerance factor (t) is a dimensionless number that is calculated from the ratio of the ionic radii:

$t={r_A+r_0 \over \sqrt{2}(r_B+r_0)}$
rA is the radius of the A-cation.	rB is the radius of the B-cation.	r0 is the radius of the anion (usually oxygen).

In an ideal cubic perovskite structure the axis of the unit cell (a) can be described with the following equation:

$a=\sqrt{2}(r_A+r_0)=2(r_B+r_0)$
rA is the radius of the A-cation.	rB is the radius of the B-cation.	r0 is the radius of the anion (usually oxygen).

Perovskite structure

The perovskite structure has the following tolerance factors (t):

Goldschmidt tolerance factor (t)	Structure	Explanation	Example	Example lattice
>1	Hexagonal	A ion too big or B ion too small.	BaNiO3	-
0.9-1	Cubic	A and B ions have ideal size.	SrTiO3 BaTiO3	300px
0.71 - 0.9	Orthorhombic/Rhombohedral	A ions too small to fit into B ion interstices.	GdFeO3 (Orthorhombic) CaTiO3 (Orthorhombic)	300px
<0.71	Different structures	A ions and B have similar ionic radii.	Ilmenite, FeTiO3 (Trigonal)	-

See also

• Goldschmidt classification
• Victor Goldschmidt

External links

• Perovskite Page at the NCSU